Abstract. We test the method of
Lockwood et al. [1999] for deriving the coronal source flux from the
geomagnetic aa index and show it to be accurate to within 12% for
annual means and 4.5% for averages over a sunspot cycle. Using data from four
solar constant monitors during 1981-1995, we find a linear relationship
between this magnetic flux and the total solar irradiance. From this
correlation, we show that the 131% rise in the mean coronal source field over
the interval 1901-1995 corresponds to a rise in the average total solar
irradiance of DI = 1.65 ±
0.23 Wm-2.

The coronal source flux

The coronal source surface is
where the solar magnetic field becomes approximately radial and lies at a
heliocentric distance of about 2.5 solar radii [Wang and Sheeley,
1995]. The total magnetic flux leaving the sun, and thereby entering the
heliosphere by threading this surface, is the coronal source flux,
Fs. Lockwood et al. [1999] have developed a method
for estimating annual means of the magnitude of the interplanetary magnetic
field, Bsw, from the aa geomagnetic index
[Mayaud, 1972]. This exploits two strong and extremely significant
correlations between the IMF, the solar wind and the aa index
[Stamper et al., 1999; Lockwood et al., 1999]. The annual means
of the components of the IMF at Earth are well described by Parker spiral
theory [Gazis, 1996; Stamper et al., 1999] giving a correlation
between Bsw and the IMF radial component, Br
. The Ulysses spacecraft has shown that variations in
Br with heliographic latitude are small [Balogh et
al., 1995], and so the total flux threading the source surface is
Fs = (|Br| ´ 4pR12)/2, where R1 = 1 AU. (The
factor of a half arises because half the flux is toward the sun, half is
away). In order to compute the annual means of Fs,
Lockwood et al. [1999] derived the necessary exponents and coefficients
from correlations between data from last three solar cycles (20-22). However,
there are uncertainties concerning the calibration of the early interplanetary
measurements [Gazis, 1996], particularly for the solar wind
concentration, Nsw. Consequently, a slightly different
approach has been adopted here. All the exponents and coefficients have been
derived using data from cycles 21 and 22 only. The predictions for cycle 20
have then been compared with the IMF observations. Thus the cycle 20 IMF data
provide a fully independent test of the method.

Figure 1.Annual means of the
coronal source magnetic flux. Those derived from the aa index,
Fs , are shown by the area shaded grey, whereas those from
near-Earth measurements of the IMF during solar cycles 20-22,
Fso , are shown by the thick line. The darker shaded area
shows the variation of the smoothed sunspot number.

Figure 1. shows the variation of coronal source flux
between 1868 and 1995, derived in this way from the aa index
(Fs, shaded grey). The solid line shows the value
Fso that is derived for the last three solar cycles from the
annual means of the observed radial IMF component, Br. The
darker-shaded region shows the sunspot number variation for comparison. Table 1 shows the RMS deviation of Fs
from Fso is similar for all three cycles
(respectively, 12%, 11% and 10% of <Fso> for cycles
20, 21 and 22). Thus the method has reproduced the variation in annual means
well, despite the fact that cycle 20 is unusual and different from cycles 21
and 22 in many ways. Table 1 also gives the
minimum-to-minimum averages, <Fs> and
<Fso>, for each sunspot cycle. It can be seen that the
error em in
<Fs> is only 1.5% for the test data, rather better
than the 4.5% for cycle 21, one of the fitted cycles. (Note that the largest
contributor to these small uncertainties is invariably the correlation between
Bsw and Br). Thus the method has
successfully extrapolated from cycles 21 and 22 to cycle 20. This means we can
apply the method to all the aa data, back to 1868, with considerable
confidence. Note that Figure 1 is not significantly
different from the results of Lockwood et al. [1999] who used
correlations based on data from all three solar cycles.

Figure 1 shows that Fs peaks shortly after the
maximum of each sunspot cycle, at about the time that the polarity of the
solar field flips. In addition to the solar cycle variation, there has been a
persistent rise in Fs since the turn of the century. The
rise is by a factor of 41% since 1964 and 131% since 1901 [Lockwood et
al., 1999]. We note that the rise in recent cycles can also be seen in the
Fso and Br data [Stamper et al.,
1999] and in the results of Wang and Sheeley [1995], who mapped the
observed photospheric field to the coronal source surface and 1AU using an
improved allowance for magnetograph saturation effects.

Table 1.Comparison of Fso
observed from IMF and Fs estimated using the aa index
for solar cycles 20, 21 and 22

Solar Cycle Number

Fitted

or test

data?

<Fso>

(1014 Wb)

<Fs>

(1014 Wb)

<Fso> -
<Fs>

(1014 Wb)

% error,

e
m = (100/<Fso>)

´
|<Fso> - <Fs>|

<(Fso
-Fs)2>1/2

(1014 Wb)

% error,

e
= 100/<Fso>)

´
<(Fso -Fs)2>1/2

20

Test

4.0881

4.0253

0.0628

1.54

0.4930

12.1

21

Fitted

4.9555

4.7316

0.2239

4.52

0.5527

11.2

22

Fitted

5.0685

5.1087

-0.0402

0.79

0.4855

9.6

Total Solar Irradiance

The output of the sun has been monitored
since 1980 by a variety of instruments, specifically: the Earth Radiation
Budget (ERB) instrument on the Nimbus spacecraft; the Active Cavity Radiometer
Irradiance Monitor (ACRIM1) on Solar Maximum Mission (SMM); the Earth
Radiation Budget Satellite (ERBS); and ACRIM2 on the Upper Atmosphere Research
Satellite (UARS). Of these, the two ACRIM experiments have been able to
monitor the degradation of their own sensors, which occurs most rapidly early
in the lifetime of the instruments. These observations show there is a solar
cycle variation in the total solar irradiance, I, of about 0.1%, but
the different instruments give different absolute values of I because
they are not consistently calibrated [see Willson, 1995]. There are two
contributions to the solar cycle variation of I: sunspots are cooler
darker regions of the photosphere, but their effects are outweighed by
associated brightenings like faculae [Lean et al., 1995]. Both sunspot
darkening and facular brightening are magnetic phenomena. If the variation in
Fs reflects changes in the solar dynamo, we would expect
associated changes in the surface field in regions of closed flux where
sunspots and faculae form. Thus we might expect some relationship between
I and Fs.

Figure 2 shows annual means of the total solar
irradiance measured by the four instruments as a function of the simultaneous
Fs, as deduced from the aa index data. To eliminate
some shorter period fluctuations, we use 3-year running means of the annual
Fs values. In each case, there is clear dependence but the
difference in the sensitivity and offset of each instrument is
apparent. Least-squares linear fits for each data set are also shown in Figure 2: the slope s, intercept (at Fs
= 0) c, correlation coefficient r, and significance level
of each fit are given in Table 2. The worst agreement is for the early data
points, for 1979 and 1980. In the case of the ERB data, this is likely to be
due, at least in part, to the rapid early degradation of the sensors and these
data have been omitted. However, higher-than-expected I is also seen by
ACRIM2 in 1980 (the point labelled in Figure 2). This is also true for the
reconstructed I produced by Lean et al. [1995]. Because it was
obtained by a self-monitoring instrument, this data point has been included in
the present study; however, its inclusion was found not to introduce any
significant change to the regressions values derived.

The key parameter for extrapolation of
I to earlier times is the slope of the regression line s. The
average s from the four instruments, <s>, is taken to have
an error of its standard deviation ss, giving a possible range of values
between smin = <s> - ss and smax =
<s> + ss. The ACRIM2
and the ERBS values lie close to, but just outside, this range. Given that
the instrumental accuracy of ACRIM2 should be the highest of the four
instruments, the best estimate of the slope may well be nearer
smax than <s>. The ACRIM2 data also give the
highest correlation coefficient; however, because this data sequence is
relatively short, this correlation has the lowest statistical significance
(which, nevertheless, exceeds 99%). For each of the instruments, neither the
systematic offset nor the sensitivity is accurately known. We here combine the
data from the different instruments by adopting a value for s and then
evaluating a mean c for this s and hence the systematic offsets
for each instrument. This was done for slopes of smin ,
<s> and smax; the results for the average slope
<s> are shown in Figure 3. Table 3 gives the offset, d, and sensitivity ratio,
g, for our inter-calibration of the various solar output
monitors. Values are broadly consistent with the ratios obtained by
Willson [1995].

The solid line in Figure 3 is the regression to
the full dataset, inter-calibrated in this way. The correlation coefficient
for the combined data is given in Table 2 and is
comparable to those for the individual datasets; however, the significance
level is higher because of the greater number of samples. We note that this
correlation between Fs and I offers an explanation of
the link between the aa index and global surface temperature on Earth
[Cliver et al., 1998].

Figure 3. Same as figure 2, with
data inter-calibrated using the average slope <s> of the
regression lines (see Table 3).

Table 3. Calibration of total
solar irradiance I for the observed values Io from
the various solar monitors,

where (I -1365) = g(Io -1365) + d
and both I and Ioare in units of W
m-2

TSI Monitor

<s>

smin

smax

g

d

g

d

g

d

Nimbus/ERB

0.9832

4.2934

0.8475

3.3628

1.1188

5.2240

SMM/ACRIM1

0.9865

-0.0331

0.8503

-0.3666

1.1226

0.3003

ERBS

1.2648

-2.1035

1.0903

-2.1512

1.4394

-2.0557

UARS/ACRIM2

0.8485

0.3260

0.7314

-0.0570

0.9656

0.7091

Long-term drift in total solar irradiance

We assume that the relation between I and Fs,
as revealed using data from solar cycles 21 and 22 in the previous section, is
valid at all times. We can then use the regressions given in Table 2 to extrapolate back to 1868. Using the mean slope
for the four instruments, <s> , yields the variation shown in Figure 4. Also shown is the 11-year running mean,
I11, which reveals a general upward trend. Given than the
heat capacity of the oceans will smooth out most of the effects of variations
in I on the timescales of the solar cycle [Wigley and Raper,
1990], these smoothed variations are most relevant to global temperature
change. The form of these curves is the same for any s, but the
amplitude of the solar cycle oscillations and of the long-term drift increases
with s. This is illustrated by Figure 5 which
shows I11 for smin , <s>
and smax. Also shown in both Figures 4 and 5 are the values
estimated by Lean et al. [1995]. The agreement between the forms of the
two extrapolations is remarkably close, considering Lean et al used sunspot
numbers to estimate sunspot darkening and facular brightening; whereas we have
used an entirely independent set of measurements, namely the aa
geomagnetic index. There is a tendency for Lean et al.ís values to be lower
in even-numbered cycles, particularly early in the century. For slopes
smin <s> and smax, we find
that the average total solar irradiance I11 increased by
DI = 1.65 ± 0.23 Wm-2 in the interval 1901-1995, up
to the value of 1367.6 Wm-2 . The lowest value of this range
(DI = 1.420 Wm-2) is a 0.10%
change, whereas the largest (DI = 1.875
Wm-2) is a 0.14% change. Lean et al. derived a slightly larger
change of DI = 2.106 Wm-2
over the same interval. The most precise instrument, ACRIM2, gives a slope
sA2 that is close to smax, and thus the
estimate by Lean et al. for the magnitude in the upward drift in I is
in close agreement with the most likely value found here.

Figure 4. The variation of the
inferred total solar irradiance I (solid line) and its 11-year running
mean I11 (dashed line), deduced using the mean of the
regression slopes, <s>. The dot-dash line is from Lean et
al. [1995].

Figure 5. The variations of the
11-year running means of the inferred total solar irradiance
I11, for the minimum, mean and maximum regression slopes
smin , <s> and smax. The
dashed curve labelled LEA is from Lean et al. [1995].

The mean rise in Fs (and Fso) over
last three solar cycles is at a rate of 0.5 ´
1014 Wb per decade and using smin ,
<s> and smax from our regression analysis gives
a rate of increase in I of 0.25± 0.4
Wm-2 per decade. We can compare this range with the estimates made
from inter-calibrated measurements during the minima at the start of cycles 22
and 23 by Willson [1995]. He reported 0.50 and 0.37 Wm-2 per
decade for ACRIM1/2 and ERBS, respectively. Thus our estimates of the recent
of rise in I are comparable with, but somewhat smaller than, those by
Willson.

The rise in I reported here DI is significant, giving a rise in the radiative forcing at the top of
the atmosphere of DQ = DI(1-a)/4 » 0.29± 0.04
Wm-2, where a is the Earthís albedo. This is comparable
with the 0.3 Wm-2 estimated by the Intergovernmental Panel on
Climate Change (IPCC) for the same interval [Schimel et al.,
1996]. Given that the IPCC estimate that the effect of anthropogenic
greenhouse CO2 is equivalent to 1.5 Wm-2, the change in
I shown in Figure 5 implies a significant role
for solar forcing of terrestrial climate change, as has also been suggested by
a number of other recent studies [e.g. Lean et al., 1995; Cliver et
al., 1998]. The effect on global mean surface temperatures will be
complex because the change in I will be made up of contributions that
are much stronger at some wavelengths (for example UV) than at others and
because a variety of other effects (for example, changes in anthropogenic
greenhouse gases, tropospheric sulphate aerosols and volcanic dust in the
stratosphere) will also be active and will interact with each other in complex
feedback loops [Rind and Overpeck, 1993]. We use the simple
relationship D Ti = -200.59 +
0.1466´I , where D Ti is the inferred temperature change
(in ° C) relative to the mean observed value
during solar cycle 11 [from Lean et al., 1995 and Rind and
Overpeck, 1993], to infer rises of 0.21, 0.24 and 0.28° C for s of, respectively, smin
, <s> and smax. This should be compared
to a rise in the global mean observed temperature D To of 0.66° C over the same interval [Parker et al.,
1994]. The net trend in D To over
the period 1870-1910 is not significantly inconsistent with the variation of
the inferred solar output. On the other hand, the change in solar luminosity
can account for only 52% of the rise in D
To over the period 1910-1960 but just 31% of the rapid rise in
D To over 1970-present. In the
interval covered by Figure 5, CO2 in the
atmosphere increased from 280 to 355 ppmv. The implications are that that the
onset of a man-made contribution to global warming was disguised by the rise
in the solar constant and that the anthropogenic effect may have a later, but
steeper, onset. This delay may have been due to an increase in the albedo
a caused by aerosols or to reduced radiative forcing DQ due to ozone depletion. There is other
evidence to support this view of the role of solar forcing: for example, the
decrease in global temperatures during about 1950-1965 was at a time when the
concentration of greenhouse gasses was increasing [Friis-Christensen and
Lassen, 1991] and the inferred decrease in solar luminosity may help
explain this. Furthermore, the present upward trend in global temperatures
commenced before significant burning of fossil fuels [Bradley and
Jones, 1993] and there is some evidence that temperatures have been as
high in past epochs as they are now [see Cliver et al., 1998].
Recently, Tett et al. [1999] have used a set of simulations made by a
coupled atmosphere-ocean global circulation model to deduce a shift from solar
forcing to anthropogenic effects as this century has progressed.

Acknowledgements This work made
use of databases and systems of the World Data Centre WDC-C1 at RAL. We thank
M. Wild who is responsible for the WDC facility and the UK Particle Physics
and Astronomy Research Council who have funded both this research and the
facility. We thank Dr Judith Lean for the Lean et al. TSI reconstruction data
and the many scientists who have contributed data to the WDC system.